With this particularly special matrix, you can observe that if you look at a particular row $r$, then it is the average of row $r - i$ and row $r + i$ (provided $r - i \geq 1$ and $r + i \leq 5$, so that the rows we're talking about are actually there). More generally, though not necessary, is that row $a$ + row $d$ = row $b$ + row $c$ for $1 \leq a \leq b \leq c \leq d \leq 5$ such that $a + d = b + c$.
This tells you that all the intermediate rows are linear combinations of the first and last row, which are visibly linearly independent since neither is a multiple of the other. Hence, $\text{rank}(A) = 2$.